A graph translation moves every point the same distance, so track the parabola’s lowest point, called its vertex. Graph the given function in Desmos and click that point. Move it as directed, then write the new equation in vertex form, , where is the new vertex. Watch the inside sign: it looks opposite to the horizontal move.
Hints
- Hint 1
The vertex is this parabola’s lowest point. Type the given function in Desmos and click that point. Which input makes the squared part equal to zero?
- Hint 2
A translation moves the whole graph without changing its shape. Apply the left move to the vertex’s -coordinate, then apply the upward move to its -coordinate.
- Hint 3
In vertex form, has vertex . Use the point you found as . The minus sign inside means the square becomes zero when .
Step-by-step
Move the vertex
Step 1Find the original vertex
Type into Desmos and click its lowest point. Desmos labels the vertex . The square is zero at , so the graph can go no lower.
- Step 2
Move the vertex left
Moving left subtracts from the vertex’s -coordinate but leaves its height at : . So the vertex moves to .
- Step 3
Move the vertex up
Moving up adds to the vertex’s -coordinate but leaves its -coordinate at : . So the new vertex is .
- Step 4
Write the translated equation
The parabola keeps its shape. In vertex form, has vertex because the square reaches its lowest value, , when . The inside expression must be so it becomes zero at the new vertex’s -coordinate, . Use : . This defines the translated graph. Choice D.